I've thought about the Probe before, and here's my take on what the unit should look like:

First off, being a worker unit it'd probably have 1 Power (going along with the SCV and MULE). Either 2-1-5 or 3-1-4 would probably look right to me. I was thinking one of its abilities would have to be to construct something, and I think a Photon Cannon that periodically created dodgeballs for you and your allies would be really cool. It'd be a high-priority target for your enemies since it'd be constantly be reinforcing your team with dodgeballs.

For the second ability, I was thinking it might create a wall of Pylons or something, but I like your idea of buffing an area with faster cooldowns and speed--more region-centric play is always good.

The Mothership Core ability is interesting--it's not something I've thought of before. I can imagine a slow-moving ally flying around and helping allies, with the enemy trying to hit it with lobs to bring it down. It's an interesting thought!

I actually really like that idea, though in my head I see it more like this:

Unit uses defensive ability and a small (invisible to enemy players) marker is created where the unit is. When the defensive ability is used again, it teleports instantaneously to the marker and the marker disappears.

I think that'd be an awesome ability! I can't say that the Probe is a unit I picture it on, but you can be sure to see it some time in the future!